Maxwell's equations

\(\renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\Grad}[1]{\mathbf{\text{grad}}\,{#1}} \newcommand{\Curl}[1]{\mathbf{\text{curl}}\,{#1}} \newcommand{\Div}[1]{\text{div}\,{#1}} \newcommand{\Real}[1]{\text{Re}({#1})} \newcommand{\Imag}[1]{\text{Im}({#1})} \newcommand{\pvec}[2]{{#1}\times{#2}} \newcommand{\psca}[2]{{#1}\cdot{#2}} \newcommand{\E}[1]{\,10^{#1}} \newcommand{\Ethree}{{\mathbb{E}^3}} \newcommand{\Etwo}{{\mathbb{E}^2}} \newcommand{\Units}[1]{[\mathrm{#1}]} \)Our aim is to solve numerically Maxwell's equations for macroscopic media: \begin{align} \Curl{\vec{h}} - \partial_t \vec{d} &= \vec{j} , \label{eq:ampere}\\ \Curl{\vec{e}} + \partial_t \vec{b} &= 0 , \label{eq:faraday}\\ \Div{\vec{b}} &= 0 , \label{eq:gaussm}\\ \Div{\vec{d}} &= \rho , \label{eq:gausse}\\ \vec{b} &= \mu_0 ( \vec{h} + \vec{m} ) \label{eq:bhm} , \\ \vec{d} &= \epsilon_0 \vec{e} + \vec{p} \label{eq:dep} . \end{align} Equations \eqref{eq:ampere}, \eqref{eq:faraday}, \eqref{eq:gaussm} and \eqref{eq:gausse} are respectively the generalized Ampère law, Faraday's law, the magnetic Gauss law and the electric Gauss law. The four vector fields $\vec{h}$, $\vec{e}$, $\vec{b}$, $\vec{d}$ are called the magnetic field, the electric field, the magnetic flux density and the electric flux density. Taken together, they form a mathematical representation of the same physical phenomenon: the electromagnetic field.

If your memory of vector analysis is bit foggy, now is a good time to freshen up as you will need at least a basic understanding of the gradient, divergence and curl operators to understand the electromagnetic tutorials.

The electric charge density $\rho$, the current density $\vec{j}$, the magnetization $\vec{m}$ and the electric polarization $\vec{p}$ are the source terms in these equations. Given $\rho$, $\vec{j}$, $\vec{m}$, $\vec{p}$ and proper initial values for $\vec{e}$ and $\vec{h}$ at the initial time instant $t=t_0$, the system \eqref{eq:ampere}–\eqref{eq:dep} determines $\vec{h}$, $\vec{e}$, $\vec{b}$, $\vec{d}$ for any other time instant $t$.

Note that \eqref{eq:gausse} implies, by \eqref{eq:ampere}, the equation of conservation of charge \begin{equation}\label{eq:charge} \Div{\vec{j}} + \partial_t \rho = 0 , \end{equation} so that, if $\vec{j}$ is given from the origin of time to the present, the charge can be obtained by integrating \eqref{eq:charge} with respect to time. In the same way, Gauss's law \eqref{eq:gaussm} can be deduced from \eqref{eq:faraday} if a zero divergence of the magnetic induction is initially assumed.

Constitutive laws

All the preceding equations are general, and have never been invalidated since their completion by Maxwell in the late 19\high{th} century. In vacuum, and, more generally, in systems that do not react with the electromagnetic field, we have $\vec{m}=0$ and $\vec{p}=0$. These systems are thus described by the two constants $\epsilon_0$ and $\mu_0$. In the MKSA system, $\mu_0=4\pi\E{-7}\text{H/m}$ and $\epsilon_0=1/(\mu_0 c^2)\text{F/m}$, where $c$ is the speed of light in vacuum.

FIXME: add table with units for all fields

In all other situations, when field-matter interaction occurs, $\vec{j}$, $\vec{m}$ and $\vec{p}$ are obtained by solving the equations describing the physical phenomena (mechanical, thermal, chemical, etc.) related to the dynamics of the charges involved in the interaction. Rigorously, one should then deal with the resolution of complex coupled systems. But the constitutive laws give us the means to bypass the explicit solving of these problems, by summarizing the complex interaction between the physical compartment of main interest (electromagnetism in this work) and those of secondary importance, the detailed modeling of which can be avoided. It is important to note that, even if all the constitutive relations are (sometimes rough) approximations of the physical behavior of the considered coupled systems, they often permit to describe very accurately the macroscopic behavior of the considered systems.

Ohm's law

The first constitutive law we adopt is Ohm's law, valid for conductors (where the current density is considered to be proportional to the electric field) and generators (where the source current density $\vec{j}_s$ can be considered as imposed, independently of the local electromagnetic field): \begin{equation}\label{eq:sigma} \vec{j} = \sigma \vec{e} + \vec{j}_s . \end{equation} The conductivity $\sigma$ is always positive (or equal to zero for insulators), and can be a tensor, in order to take an anisotropic behavior into account. Note that this relation is only valid for non-moving conductors: for a conductor moving at speed $\vec{v}$, \eqref{eq:sigma} becomes $\vec{j} = \sigma (\vec{e} + \pvec{\vec{v}}{\vec{b}}) + \vec{j}_s$.

Dielectric constitutive law

The second constitutive law describes the behavior of dielectric materials, stating a proportionality between the polarization and the electric field, as it would be if the charges were elastically bound, with a restoring force proportional to the electric field: \begin{equation}\label{eq:pe} \vec{p} = \chi_e \vec{e} + \vec{p}_e . \end{equation} Again, the electric susceptibility $\chi_e$ can be a tensor to describe an anisotropic behavior. A permanent polarization $\vec{p}_e$ is considered for materials exhibiting a permanent polarization independent of the electric field, such as electrets. Introducing \eqref{eq:pe} in \eqref{eq:dep}, we get \begin{align} \vec{d} & = \epsilon_0 \vec{e} + \chi_e \vec{e} + \vec{p}_e \nonumber\\ & = (\epsilon_0 + \chi_e ) \vec{e} + \vec{p}_e \nonumber\\ & = \epsilon_0 \epsilon_r \vec{e} + \vec{p}_e \nonumber\\ & = \epsilon \vec{e} + \vec{p}_e \label{eq:epsilon} , \end{align} where $\epsilon$ and $\epsilon_r=1+\chi_e/\epsilon_0$ are the electric permittivity and the relative electric permittivity of the material respectively.

Magnetic constitutive law

The third constitutive law expresses an approximate relation between the magnetization and the magnetic field in magnetic materials: \begin{equation}\label{eq:mh} \vec{m} = \chi_m \vec{h} + \vec{h}_m . \end{equation} For paramagnetic and ferromagnetic materials, the magnetic susceptibility $\chi_m$ is always positive. For diamagnetic materials the magnetic susceptibility is negative. Again, it can be a tensor to describe an anisotropic behavior. For permanent magnets~\cite{lacroux-aimants-89}, one considers a non-zero permanent magnetic field $\vec{h}_m$ supported by the magnet, and independent of the local magnetic field. Introducing \eqref{eq:mh} in \eqref{eq:bhm}, we get \begin{align} \vec{b} & = \mu_0 ( \vec{h} + \chi_m \vec{h} + \vec{h}_m ) \nonumber\\ & = \mu_0 ( 1 + \chi_m ) \vec{h} + \mu_0 \vec{h}_m \nonumber\\ & = \mu_0 \mu_r \vec{h} + \mu_0 \vec{h}_m \nonumber\\ & = \mu \vec{h} + \mu_0 \vec{h}_m \label{eq:mu} , \end{align} where $\mu$ and $\mu_r=1+\chi_m$ are the magnetic permeability and the relative magnetic permeability of the material respectively.

In the simplest modeling option, the material characteristics $\sigma$, $\epsilon$ and $\mu$ involved in \eqref{eq:sigma}, \eqref{eq:epsilon} and \eqref{eq:mu} are considered as constants. This situation is the linear case without memory.

Lorentz Force

FIXME: todo

Time integration

Two main strategies for the time integration of the equations can be used. For a classical time domain analysis, appropriate initial conditions must be provided. But if the system is fed by a sinusoidal excitation and if its response is linear (which is the case if all operators and the material characteristics are linear), the problem can also be solved in the frequency domain. For a sinusoidal variation of angular frequency $\omega$, any field can then be described as \begin{equation} f(\vec{x},t) = f_m(\vec{x}) \cos(\omega t+\varphi(\vec{x})), \end{equation} where $\varphi(\vec{x})$ is a phase angle (expressed in radians) which can depend on the position. The harmonic approach then consists in defining this physical field as the real part of a complex field, i.e.: \begin{equation}\label{eq:cplx} f(\vec{x},t) = \Real{f_m(\vec{x}) e^{i(\omega t+\varphi(\vec{x}))}} = \Real{f_p(\vec{x}) e^{i\omega t}} , \end{equation} where $i=\sqrt{-1}$ denotes the imaginary unit. The complex field \begin{equation} f_p(\vec{x}) = f_m(\vec{x}) e^{i\varphi(\vec{x})} = f_r(\vec{x}) + i f_i(\vec{x}) \end{equation} appearing in \eqref{eq:cplx} is called a phasor, $f_r(\vec{x})$ and $f_i(\vec{x})$ being its real and imaginary parts respectively. If all physical fields are expressed as in \eqref{eq:cplx}, their substitution in the equations of the system leads to complex equations, the unknowns of which are phasors. Through \eqref{eq:cplx}, the time derivative operator becomes a product by the factor $i\omega$. In particular, Maxwell's equations \eqref{eq:ampere}–\eqref{eq:gausse} in harmonic regime become \begin{gather} \Curl{\vec{h}} - i\omega \vec{d} = \vec{j} \label{eq:ampere:cplx} , \\ \Curl{\vec{e}} + i\omega \vec{b} = 0 \label{eq:faraday:cplx} , \\ \Div{\vec{b}} = 0 \label{eq:gaussm:cplx} , \\ \Div{\vec{d}} = \rho \label{eq:gausse:cplx} , \end{gather} where all the fields are now phasors.